The Hovey–Strickland and Chai conjectures

Claims Chai's invariant-ideal conjecture at every prime and height, and with it the Hovey–Strickland classification.

1 paper September 24, 2026 Explained by AI

Chromatic homotopy theory splits stable homotopy theory into layers indexed by a height n. Hovey and Strickland conjectured that one layer's dualizable objects have exactly n+2 thick tensor ideals, a list of closed families. Barthel, Heard and Naumann showed this would follow from Chai's conjecture about symmetric ideals in a ring of power series. This paper claims Chai's conjecture at every prime and height. So it also claims the Hovey–Strickland count.

First step

1999

Mark Hovey and Neil Strickland, Morava K-theories and localisation

Significance

Major

It claims two linked conjectures at every height, where the spectral one was known only at height two.

Trust

Claimed, not checked

Field

Topology

1 paper

The explanation

Start with a ring of power series. At height n = 3 it is E₀ = W(F_{p³})[[u₁, u₂]]. Here W(F_{p³}) is a ring of p-adic integers built from a finite field with p³ elements. This ring has many prime ideals, the basic building blocks of its ideals. But a large symmetry group acts on it, called the Morava stabilizer group. Only some ideals stay fixed under that action. There are obvious ones, the height ideals: 0, (p), (p, u₁) and (p, u₁, u₂). That is n + 1 = 4 fixed primes. Adding the whole ring gives 5 fixed radical ideals, which is n + 2.

Chai proposed that the fixed loci should be exactly these height lociSource 1. The paper claims that the fixed proper prime ideals are exactly 0 and the height idealsSource 2. This holds for every open subgroup of the stabilizer, which extends the earlier formulationSource 3. So the paper claims Chai's conjecture in its standard formSource 4. The hard part is the layers in characteristic p. There, small symmetry directions alone do not control the fixed equationsSource 5.

Why does this matter for topology? Hopkins and Smith classified the thick subcategories of finite spectra by chromatic typeSource 6. A thick subcategory is a family closed under the basic building moves. Hovey and Strickland proposed the same kind of list for dualizable K(n)-local spectraSource 7. Dualizable objects have a well-behaved dual. This class is larger than the compact objectsSource 8. Barthel, Heard and Naumann proved this list at height two. They also showed that Chai's conjecture would give it at every heightSource 9.

With that implication, the paper claims the Hovey–Strickland conjecture. The dualizable K(n)-local category has exactly n + 2 thick tensor ideals. Its Balmer spectrum, the space of prime ideals, is a chain of n + 1 pointsSource 10. At height 3 that is 5 ideals and 4 points. This matches the 4 fixed primes in the example above.

How sure is this? There is no Lean check. A computer proof checker has not looked at any part of it. The proof uses deep tools from p-adic geometry. The topological part relies on the earlier Barthel–Heard–Naumann implication. No specialist review of the paper has been published.

What it doesn't show. This does not classify ideals in the larger category of all K(n)-local spectra, only the dualizable ones.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if the proof holds. The topological conclusion depends on earlier published work. There is no Lean check and no published specialist review.

Words used

Height: The index n of a layer in chromatic homotopy theory.
Prime ideal: An ideal that plays the role of a prime number in a ring.
Radical ideal: An ideal that contains every element some power of which lies in it.
Morava stabilizer group: A large symmetry group that acts on the Lubin–Tate ring.
Thick tensor ideal: A family of objects closed under the basic building moves and under tensoring with anything.
Dualizable: An object with a well-behaved dual, so that maps out of it match maps into its dual.
Balmer spectrum: The space of prime thick tensor ideals of a tensor category.

The original paper

Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra (PDF, opens in a new tab)
September 24, 2026

All quoted lines (10)

Chai studied the action on the closed fiber of Lubin–Tate space and proposed that invariant irreducible formal loci should be height loci.
Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra Back to text
The proper prime ideals of $E_0$ stabilized by $U$ are precisely $0,\ I_1,\ldots,I_n.$
Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra Back to text
This proves the invariant-ideal assertion called Chai's Hope in the finite-residue-field formulation of, and extends that formulation from the full stabilizer to every open subgroup.
Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra Back to text
This proves Chai's invariant-ideal conjecture in the standard finite-residue-field formulation.
Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra Back to text
The main issue addressed here is the characteristic- $p$ height strata, where infinitesimal stabilizer directions alone do not give the required control of invariant equations.
Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra Back to text
The thick subcategory theorem of Hopkins and Smith classifies thick subcategories of finite spectra by chromatic type, while their periodicity theorem supplies the corresponding self-maps.
Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra Back to text
Hovey and Strickland proposed the dualizable $K(n)$ -local counterpart in their study of Morava $K$ -theory and localization.
Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra Back to text
Dualizability is essential here: it is more general than compactness in the $K(n)$ -local category.
Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra Back to text
Barthel, Heard, and Naumann proved the Hovey–Strickland conjecture at height two for every prime and established the general implication from Chai's invariant-ideal assertion to the spectral classification.
Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra Back to text
It also proves the Hovey–Strickland conjecture: the dualizable $K(n)$ -local category has exactly $n+2$ thick tensor ideals, and its Balmer spectrum is a chain of $n+1$ points.
Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra Back to text

Other sources

openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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